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Applied Business StatisticsApplied Business StatisticsCase studiesCase studies

Credit riskCredit risk

Mauro BufanoMauro Bufano

Risk Management – Risk Management – BancaBanca Mediolanum Spa Mediolanum Spa

Credit risk – what is?Credit risk – what is?

Credit risk can be defined as “the possibility that an Credit risk can be defined as “the possibility that an unexpected change in a counterparty’s creditworthiness unexpected change in a counterparty’s creditworthiness may generate a corresponding unexpected change in may generate a corresponding unexpected change in the market value of the associated credit exposure”the market value of the associated credit exposure”11

Therefore credit risk doesn’t take into account only Therefore credit risk doesn’t take into account only counterparty’s failure, but any event that could reduce counterparty’s failure, but any event that could reduce the value of the exposure (e.g., a decrease in the the value of the exposure (e.g., a decrease in the guarantees)guarantees)

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Definition of defaultDefinition of default

Generally, there is a default when we observe (Moody’s Generally, there is a default when we observe (Moody’s definition):definition):

missed payment of a financial obligation (e.g. a missed payment of a financial obligation (e.g. a mortgage instalment) more than 90 daysmortgage instalment) more than 90 days

bankruptcybankruptcy liquidationliquidation debt restructuring debt restructuring

Therefore, there are different types of default. Every bank Therefore, there are different types of default. Every bank can anyway fix different limits to evaluate a credit can anyway fix different limits to evaluate a credit exposure as “default”exposure as “default”

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Credit risk – the main Credit risk – the main componentscomponents

If we analyze credit risk by single exposure, we have 3 If we analyze credit risk by single exposure, we have 3 different components (or parameters) to be estimated:different components (or parameters) to be estimated: Exposure at defaultExposure at default: (EAD): : (EAD): it’s the expected value of it’s the expected value of

the exposure in the moment of defaultthe exposure in the moment of default Probability of default Probability of default (PD): (PD): it’s the probability that the it’s the probability that the

borrower will not repay (entirely or in part) the loan borrower will not repay (entirely or in part) the loan within a given timewithin a given time

Loss given default Loss given default (LGD): (LGD): it’s the expected loss rate in it’s the expected loss rate in case of default, i.e. the percentage of exposure that a case of default, i.e. the percentage of exposure that a bank will be unable to recover it depends bank will be unable to recover it depends on the guarantees (e.g. the value of the house in on the guarantees (e.g. the value of the house in mortgage loans)mortgage loans)

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Credit riskCredit riskTherefore, credit risk comprises the following main risks:Therefore, credit risk comprises the following main risks:

• Default risk:Default risk: it’s the risk connected to the default of it’s the risk connected to the default of the counterpartythe counterparty

• Migration risk: Migration risk: it’s the risk connected with a it’s the risk connected with a deterioration in the counterparty’s creditworthinessdeterioration in the counterparty’s creditworthiness

• Spread risk:Spread risk: it’s the risk that an increasing probability it’s the risk that an increasing probability of default (or reduced guarantees) cause a higher of default (or reduced guarantees) cause a higher credit spread on the market and lower values of credit spread on the market and lower values of issuers’ bondsissuers’ bonds

• Recovery risk:Recovery risk: it’s the risk that the guarantees of the it’s the risk that the guarantees of the insolvent counterparty are less than expected and/or insolvent counterparty are less than expected and/or the recovery process takes longer than expectedthe recovery process takes longer than expected

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Exposure at defaultExposure at default EAD is sometimes deterministic and direct observable EAD is sometimes deterministic and direct observable

(e.g. for a bond it coincides with its market value) but we (e.g. for a bond it coincides with its market value) but we have many cases in which it has to be estimated:have many cases in which it has to be estimated:

a.a. Because of the uncertainty in the amount of funds Because of the uncertainty in the amount of funds taken by the borrower (e.g., a credit line)taken by the borrower (e.g., a credit line)

b.b. Because of the uncertainty of the residual debt at Because of the uncertainty of the residual debt at the moment of default (e.g. loans to be repaid in the moment of default (e.g. loans to be repaid in instalments)instalments)

c.c. Because of the uncertainty in the market value of Because of the uncertainty in the market value of some financial instruments (e.g. “over the counter” some financial instruments (e.g. “over the counter” derivatives)derivatives)

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In estimating EAD generally we fix a “drawn” portion In estimating EAD generally we fix a “drawn” portion (DP, the part “used” in a credit line) and the “undrawn” (DP, the part “used” in a credit line) and the “undrawn” portion (UP, the difference between the credit line and portion (UP, the difference between the credit line and the amount used)the amount used)

The Exposure of default is given by:The Exposure of default is given by:

Where CCF is the “credit conversion factor”. This formula Where CCF is the “credit conversion factor”. This formula takes into account the fact that “stressed” borrowers tend takes into account the fact that “stressed” borrowers tend to “use” more their credit line, because of lack of liquidityto “use” more their credit line, because of lack of liquidity

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Exposure at defaultExposure at default

CCFUPDPEAD

From PD to ratingsFrom PD to ratings

Generally banks don’t estimate a single PD for each Generally banks don’t estimate a single PD for each counterparty, but they classify the borrowers with an counterparty, but they classify the borrowers with an ordinal measure: the ordinal measure: the ratingrating

A credit rating represents an evaluation of credit A credit rating represents an evaluation of credit worthiness of a borrowerworthiness of a borrower

Ratings take into account not only quantitative data (e.g. Ratings take into account not only quantitative data (e.g. taken from the balance sheet) but also qualitative taken from the balance sheet) but also qualitative information of a counterpartyinformation of a counterparty

Biggest company are rated by agencies (S&P, Moody’s, Biggest company are rated by agencies (S&P, Moody’s, Fitch), while smaller companies or customers are rated Fitch), while smaller companies or customers are rated by each banks with their by each banks with their internal modelsinternal models

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Rating scale – an exampleRating scale – an exampleHere is reported as example of rating scale (Standard & Here is reported as example of rating scale (Standard & Poor’s)Poor’s)

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Rating class

AAA

AA

A

BBB

BB

B

CCC

CC

C

D

Investment grade (“good” borrowers)

Speculative grade (“bad” borrowers)

Defaulted borrowers

Rating assignmentRating assignment Here is reported an example of workflow for the rating Here is reported an example of workflow for the rating

assignmentassignment

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Data collection of the borrowers (e.g. balance sheet data, residential data etc.)

Selection of the leading indicators for credit worthiness

Transformation of the leading indicators into a quantitative “score”

Mapping of the score into a

rating

Selection of variablesSelection of variables The first step in the construction of a statistical model for The first step in the construction of a statistical model for

the rating assignment is the definition of the the rating assignment is the definition of the methodology for rating assignment. It could be either:methodology for rating assignment. It could be either:

JudgmentalJudgmental: the rating is assigned basing on the opinion of the : the rating is assigned basing on the opinion of the analysts – generally used for large corporate and banksanalysts – generally used for large corporate and banks

ScoringScoring: : the borrower receives a the borrower receives a scorescore, obtained with statistical , obtained with statistical methods – generally used for retail exposuresmethods – generally used for retail exposures

MixedMixed: the rating is assigned using either judgmental and : the rating is assigned using either judgmental and scoring techniquesscoring techniques

Then, it’s necessary to build the dataset, both with Then, it’s necessary to build the dataset, both with qualitative and quantitative dataqualitative and quantitative data

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Selection of variablesSelection of variables In building the dataset, we must take into account:In building the dataset, we must take into account:

• The nature of the data and the historical deepnessThe nature of the data and the historical deepness• Missing data (eventually substituted by the mean or median Missing data (eventually substituted by the mean or median

value)value)• Analysis of duplicate cases (e.g. one customer has several Analysis of duplicate cases (e.g. one customer has several

loans) loans) • Percentage of “good” and “bad” in the datasetPercentage of “good” and “bad” in the dataset• Heterogeneity of the dataHeterogeneity of the data

Once the dataset is complete, the candidate variables Once the dataset is complete, the candidate variables must have:must have:

• economic senseeconomic sense • discriminatory powerdiscriminatory power• low correlation with other variableslow correlation with other variables

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Examples of variablesExamples of variables

Examples of variables that could be included in the Examples of variables that could be included in the dataset (e.g. for corporates)dataset (e.g. for corporates)

Earnings before Interest and Taxes (EBIT)Earnings before Interest and Taxes (EBIT) Return on equity (ROE)Return on equity (ROE) Total debt / CapitalTotal debt / Capital Operating income /salesOperating income /sales Long term debt / capitalLong term debt / capital Operating cash flows / Total debtOperating cash flows / Total debt

Generally ratios are preferred to absolute values, Generally ratios are preferred to absolute values, because in this way we have independence from the because in this way we have independence from the size of the companysize of the company

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Logistic regressionLogistic regression

The most used model in credit risk analysis is the logistic The most used model in credit risk analysis is the logistic regression. The logistic function is defined as followsregression. The logistic function is defined as follows

With With zz being defined as: being defined as:

where where ββ00 is the intercept and  is the intercept and ββ11… β… βkk are the regression are the regression

coefficients of coefficients of xx11…x…xkk respectively respectively

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)exp(1

1)(

zzf

kk xxz ...110

Logistic regressionLogistic regressionOne of the main advantages of logistic regression is represented by the One of the main advantages of logistic regression is represented by the

fact that fact that  it can take as an input any value from negative infinity to  it can take as an input any value from negative infinity to positive infinity, whereas the output is confined to values between 0 positive infinity, whereas the output is confined to values between 0 and 1 (see figure below).and 1 (see figure below).

Therefore, it’s a useful way of describing the relationship between one Therefore, it’s a useful way of describing the relationship between one or more independent variables (e.g., age, income, etc.) and a binary or more independent variables (e.g., age, income, etc.) and a binary response variable, expressed as a probability, that has only two response variable, expressed as a probability, that has only two possible values, such as default ("default" or "no default").possible values, such as default ("default" or "no default").

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Univariate analysisUnivariate analysisOnce we have chosen the explicative variables, we can start with the Once we have chosen the explicative variables, we can start with the

univariate analysis: it consists in building separate univariate logistic univariate analysis: it consists in building separate univariate logistic regression and evaluate, for each explicative variable, the goodness regression and evaluate, for each explicative variable, the goodness of fit and the coherence with the economic assumption (e.g. higher of fit and the coherence with the economic assumption (e.g. higher debt/equity would correspond to lower credit quality of borrowersdebt/equity would correspond to lower credit quality of borrowers

In this part of the analysis several checks must be done:In this part of the analysis several checks must be done:• Direction analysis: Direction analysis: it analyzes if the average values of the two it analyzes if the average values of the two

datasets (bads and goods) are respectively positioned as datasets (bads and goods) are respectively positioned as expectedexpected

• Test of normality: Test of normality: it can be conducted with the Kolmogorov-it can be conducted with the Kolmogorov-Smirnov testSmirnov test

• Test of homogeneity of variances: Test of homogeneity of variances: Levene’s testLevene’s test• Test of discriminatory capabilityTest of discriminatory capability• Analysis of correlations: Analysis of correlations: in order to exclude high correlated in order to exclude high correlated

explanatory variablesexplanatory variables

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Box plotsBox plots An useful instrument used to evaluate the discriminatory power of a An useful instrument used to evaluate the discriminatory power of a

variable is the box plot. It allows to compare the dispersion of the variable is the box plot. It allows to compare the dispersion of the two dataset in a graphical waytwo dataset in a graphical way

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In this case, we can see how the dataset 1 (good) differ from the dataset 2 (bad), therefore this variable has an explanatory power

2004SIZE = log-Equity

1 210

11

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13

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18

Valu

es

Column Number

2005Capital = Total Capital Ratio

1 2

10

15

20

25

30

35

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45

50

Valu

es

Column Number

In this case, instead, there is not a clear diversification between the two datasets, therefore this variable has not an explanatory power

Multivariate analysisMultivariate analysis After the univariate analysis, we can build the multifactor model. After the univariate analysis, we can build the multifactor model.

One of the most used procedures is named One of the most used procedures is named “step forward”:“step forward”: after after having chosen the having chosen the bestbest univariate model (in terms of goodness of fit, univariate model (in terms of goodness of fit, e.g. with the R-square), we proceed by adding variables to the e.g. with the R-square), we proceed by adding variables to the model one-by-one, testing the predictive power of the model with model one-by-one, testing the predictive power of the model with several tests:several tests:

Log likelihood ratio test:Log likelihood ratio test: we compare the log-likelihoods of the we compare the log-likelihoods of the two models. The variable is included if the difference is two models. The variable is included if the difference is significantsignificant

Wald statistic test:Wald statistic test: tests the significance of tests the significance of ββ Multicollinearity:Multicollinearity: excludes variables with a correlation higher of excludes variables with a correlation higher of

60%60% Box-Tidwell test:Box-Tidwell test: tests the presence of non-linear effects tests the presence of non-linear effects

between the between the yy and the and the xx

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An important passage is represented by the An important passage is represented by the back-testback-test, , that must be done that must be done out-of sample, out-of sample, in order to check the in order to check the stability of the modelstability of the model

In general, a robust model should have the following In general, a robust model should have the following characteristics:characteristics:

Credit senseCredit sense Good performances in the fit and predictive powerGood performances in the fit and predictive power Heterogeneity among explicative variablesHeterogeneity among explicative variables in order to in order to

capture different sources of riskcapture different sources of risk Performance along time Performance along time Parsimony! Parsimony! The number of variables in the models should The number of variables in the models should

be lowbe low

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Multivariate analysisMultivariate analysis

Mapping of a score into a ratingMapping of a score into a rating The output of the model is a “score”. Suppose we have a score The output of the model is a “score”. Suppose we have a score

between 0 and 100 (as an output of logistic regression). How do we between 0 and 100 (as an output of logistic regression). How do we map it into a rating? We can build grids like these (whose map it into a rating? We can build grids like these (whose performance is, anyway, to be tested)performance is, anyway, to be tested)

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Rating class Score

AAA 95-100

AA 90-95

A 80-90

BBB 70-80

BB 60-70

B 40-50

CCC 30-40

CC 20-30

C 10-20

D 0-10

Rating quantification: PD Rating quantification: PD assignmentassignment

In order to estimate expected losses/unexpected losses we have to In order to estimate expected losses/unexpected losses we have to associate to each rating class a numeric measure: the probability of associate to each rating class a numeric measure: the probability of default (PD)default (PD)

Generally, PDs are estimated through historical default ratesGenerally, PDs are estimated through historical default rates In this way we can obtain a “PD scale”, that must have the following In this way we can obtain a “PD scale”, that must have the following

characteristics:characteristics: MonotonicityMonotonicity: upper rating class must have lower PD: upper rating class must have lower PD Positivity:Positivity: the PD must be strictly positive also in upper rating the PD must be strictly positive also in upper rating

classes (i.e. the minimum PD required by Basel II is 0,03%)classes (i.e. the minimum PD required by Basel II is 0,03%) Reasonable:Reasonable: not too far from reality not too far from reality Conservative:Conservative: taking into account stressed scenarios taking into account stressed scenarios Stability:Stability: the change in credit-worthiness of a particular subject the change in credit-worthiness of a particular subject

must reflect in a rating-change, but the PD associated to a rating must reflect in a rating-change, but the PD associated to a rating class should be as much as stable through timeclass should be as much as stable through time

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An example of PD scaleAn example of PD scale

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Let’s now consider historical average default rates per rating class in the period 1981-2008, as reported by Standard & Poor’s

expy x

A possible model could be built through the “exponential fit”, following the formula below. The parameters α and β can be estimated by minimizing the sum of square of errors

Here is reported an example of exponential PD scale, with PD constrained to be greater than 0,03%Here is reported an example of exponential PD scale, with PD constrained to be greater than 0,03% In the chart is reported also the comparison with the 2008 default rate (“a stressed scenario”) in order to evaluate whether In the chart is reported also the comparison with the 2008 default rate (“a stressed scenario”) in order to evaluate whether

the model is conservativethe model is conservative

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An example of PD scaleAn example of PD scale

Rating class

Historical default rates

1981-2008

2008 default rates fitted curve

AAA 0.00% 0.00% 0.03%

AA+ 0.00% 0.00% 0.04%

AA 0.02% 0.43% 0.07%

AA- 0.03% 0.40% 0.10%

A+ 0.05% 0.31% 0.15%

A 0.06% 0.21% 0.23%

A- 0.08% 0.58% 0.34%

BBB+ 0.16% 0.18% 0.51%

BBB 0.28% 0.59% 0.76%

BBB- 0.28% 0.71% 1.13%

BB+ 0.68% 1.14% 1.70%

BB 0.89% 0.63% 2.54%

BB- 1.53% 0.63% 3.80%

B+ 2.44% 2.97% 5.69%

B 7.28% 3.29% 8.51%

B- 9.97% 7.02% 12.74%

CCC/C 22.67% 26.53% 19.08%

0%

5%

10%

15%

20%

25%

30%

AAA AA A+ A- BBB BB+ BB- B CCC/C

Defa

ult

rate

(%

)

Fitting actual default rates

Historical default rate

Fitted curve

2008 default rates

The determination of “expected The determination of “expected loss”loss”

Having estimated the EAD, the PD and the LGD for every Having estimated the EAD, the PD and the LGD for every exposure, we can determine the “expected loss” or average exposure, we can determine the “expected loss” or average loss at the single exposure level, but also of the entire loan loss at the single exposure level, but also of the entire loan portfolio (see example below)portfolio (see example below)

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Counterparty Exposure at default Rating PD LGD Expected LossA 950.000 € AAA 0,03% 20% 57 €B 550.000 € AA- 0,10% 20% 110 €C 120.000 € BBB+ 0,51% 35% 214 €D 250.000 € BB 2,54% 50% 3.175 €E 300.000 € A+ 0,15% 30% 135 €F 400.000 € A- 0,34% 25% 340 €G 500.000 € AA- 0,10% 30% 150 €H 150.000 € A+ 0,15% 50% 113 €I 320.000 € BBB- 1,13% 80% 2.893 €J 870.000 € A 0,23% 25% 500 €

Entire portfolio 4.410.000 € 7.687 €0,17%

A particular case “the shadow A particular case “the shadow rating”rating”

There some cases in which traditional credit models based on There some cases in which traditional credit models based on logistic regression aren’t fit. This happens when the sample logistic regression aren’t fit. This happens when the sample considered has experienced very few or no default (e.g. banks or considered has experienced very few or no default (e.g. banks or governments)governments)

In this case the methodology used is named “shadow rating”: the In this case the methodology used is named “shadow rating”: the yy of the regression is not anymore the status “default /no default” but of the regression is not anymore the status “default /no default” but the rating itself (generally we take as benchmark the agency rating)the rating itself (generally we take as benchmark the agency rating)

The regression method used is named “Ordered Logistic The regression method used is named “Ordered Logistic Regression” (OLR), in which we regress a latent variable (Regression” (OLR), in which we regress a latent variable (y*y*) – ) – representing a score associated to a rating class – to the explicative representing a score associated to a rating class – to the explicative variablesvariables

Then, we map the score to the estimated ratingThen, we map the score to the estimated rating

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From single exposure to a loan From single exposure to a loan portfolio portfolio

Until now, we have analyzed credit risk “stand alone”, by Until now, we have analyzed credit risk “stand alone”, by taking into account the risk associated to single taking into account the risk associated to single exposures. When we move from single exposures to a exposures. When we move from single exposures to a loan portfolio, we must consider two additional risks:loan portfolio, we must consider two additional risks:

Concentration riskConcentration risk: it’s the risk that the exposures are heavily : it’s the risk that the exposures are heavily concentrated on few counterparties, therefore an eventual concentrated on few counterparties, therefore an eventual default could cause huge losses for the bankdefault could cause huge losses for the bank

Systemic riskSystemic risk: it’s the risk due to economic downturn or industry : it’s the risk due to economic downturn or industry crisis (e.g. the 2001 crisis that affected all web companies). In crisis (e.g. the 2001 crisis that affected all web companies). In this case what matters is this case what matters is portfolio diversificationportfolio diversification and the and the correlation of the single counterparty with the risk factors.correlation of the single counterparty with the risk factors.

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Concentration riskConcentration riskConcentration risk is generally measured with the Herfindahl indexConcentration risk is generally measured with the Herfindahl index

where where wwii is the percentage exposure of the i-th counterpartyis the percentage exposure of the i-th counterparty

Let’s consider the two portfolios belowLet’s consider the two portfolios below

The two portfolios have the same PDs and LGDs, but the second one is more concentrated on the first counterparties. An eventual default of counterpart A would impact 100,000 € in the first portfolio, in the second one 500,000 €!The two portfolios have the same PDs and LGDs, but the second one is more concentrated on the first counterparties. An eventual default of counterpart A would impact 100,000 € in the first portfolio, in the second one 500,000 €!

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n

iiwH

1

2

Ctp EAD PD LGD Expected LossA 100.000 € 1,00% 20% 200 €B 100.000 € 1,00% 20% 200 €C 100.000 € 1,00% 20% 200 €D 100.000 € 1,00% 20% 200 €E 100.000 € 1,00% 20% 200 €F 100.000 € 1,00% 20% 200 €G 100.000 € 1,00% 20% 200 €H 100.000 € 1,00% 20% 200 €I 100.000 € 1,00% 20% 200 €J 100.000 € 1,00% 20% 200 €

Entire portfolio 1.000.000 € 2.000 €0,20%

10%Herfindahl Index

Ctp EAD PD LGD Expected LossA 500.000 € 1,00% 20% 1.000 €B 250.000 € 1,00% 20% 500 €C 100.000 € 1,00% 20% 200 €D 50.000 € 1,00% 20% 100 €E 50.000 € 1,00% 20% 100 €F 10.000 € 1,00% 20% 20 €G 10.000 € 1,00% 20% 20 €H 10.000 € 1,00% 20% 20 €I 10.000 € 1,00% 20% 20 €J 10.000 € 1,00% 20% 20 €

Entire portfolio 1.000.000 € 2.000 €0,20%

33%Herfindahl Index

Systemic riskSystemic risk

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Systemic risk depends on the risk factor that could cause the default of Systemic risk depends on the risk factor that could cause the default of several counterparties (e.g. In 2007 the real estate crisis in the US several counterparties (e.g. In 2007 the real estate crisis in the US caused the default of many mortgage loans)caused the default of many mortgage loans)

Therefore, it’s necessary to:Therefore, it’s necessary to:

a)a) Individuate the risk factorsIndividuate the risk factors

b)b) Estimate the correlation (default correlation)Estimate the correlation (default correlation) of each of each counterparty with the risk factorcounterparty with the risk factor

The estimation of b) is sometimes very complex, and generally The estimation of b) is sometimes very complex, and generally approximated with the asset correlation (e.g. correlation of approximated with the asset correlation (e.g. correlation of equity prices, when the counterparty is a listed company on the equity prices, when the counterparty is a listed company on the stock exchange)stock exchange)

Generally systemic risk is higher if the exposure are concentrated in Generally systemic risk is higher if the exposure are concentrated in a particular region / industry (e.g. a bank specialised in loans in a particular region / industry (e.g. a bank specialised in loans in the real estate market in a region of Italy) it’s a strong the real estate market in a region of Italy) it’s a strong argument against bank specializationargument against bank specialization

The Credit VaRThe Credit VaRAs for market risk, we can work out a measure of unexpected losses using As for market risk, we can work out a measure of unexpected losses using

the concept of Value at Risk (VaR).the concept of Value at Risk (VaR).

In this case it’s necessary to simulate the distribution of losses due to default In this case it’s necessary to simulate the distribution of losses due to default events it’s therefore defined only on the positive scale (see figure* events it’s therefore defined only on the positive scale (see figure* below).below).

**http://www.mizuho-fg.co.jp/english/company/internal/r_management/images/internal_ph03_03.gifhttp://www.mizuho-fg.co.jp/english/company/internal/r_management/images/internal_ph03_03.gif

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Example of construction of Credit VaRExample of construction of Credit VaR Let’s consider, for simplicity, a portfolio composed by N counterparties, each with a EAD, PD and LGDLet’s consider, for simplicity, a portfolio composed by N counterparties, each with a EAD, PD and LGD There is only one risk factorThere is only one risk factor Let’s suppose the asset value return of the i-th counterparty is given byLet’s suppose the asset value return of the i-th counterparty is given by

WhereWhere ρρi i is the correlation of the i-th counterparty with the common risk factor is the correlation of the i-th counterparty with the common risk factor yy εεii is the idiosyncratic factor of the i-th counterparty is the idiosyncratic factor of the i-th counterparty

A counterparty will default if its asset returns goes below a given threshold, that is supposed to be the inverse of the cumulated standard Normal A counterparty will default if its asset returns goes below a given threshold, that is supposed to be the inverse of the cumulated standard Normal distribution evaluated in the PDdistribution evaluated in the PD

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iiii yx 21

The simulation has the following steps The simulation has the following steps (“dice technique”):(“dice technique”): We extract a random number for We extract a random number for y y and for each and for each εεii For each For each xxi i , we compare, we compare the simulated number with Nthe simulated number with N-1-1(PD(PDii))

• if if xxii < < NN-1-1(PD(PDii) the i-th counterparty default and the associated loss is ) the i-th counterparty default and the associated loss is

EADEADii*LGD*LGDii

• otherwise, the i-th counterparty doesn’t default and the associated loss is otherwise, the i-th counterparty doesn’t default and the associated loss is zerozero

At the end of the simulation, we sum the losses for each At the end of the simulation, we sum the losses for each counterparty obtaining the aggregate losses for the portfoliocounterparty obtaining the aggregate losses for the portfolio

We repeat the simulation several times (e.g. 100,000) in order to We repeat the simulation several times (e.g. 100,000) in order to get a get a portfolio loss distributionportfolio loss distribution

The Credit VaR with The Credit VaR with xx confidence level is the confidence level is the x-th x-th percentile in percentile in the portfolio loss distributionthe portfolio loss distribution

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Example of construction of Credit VaRExample of construction of Credit VaR

Decomposition of Credit VaRDecomposition of Credit VaR

Credit VaR can be decomposed into three main elements:Credit VaR can be decomposed into three main elements: The The specific riskspecific risk: it’s given by the expected loss. It represent the : it’s given by the expected loss. It represent the

average riskiness of the portfolioaverage riskiness of the portfolio The The concentration riskconcentration risk: it’s given by the difference between the : it’s given by the difference between the

Credit Var (simulated without the risk factors) and the expected Credit Var (simulated without the risk factors) and the expected lossloss

The The systemic risksystemic risk: it’s given by the difference between Credit : it’s given by the difference between Credit VaR (simulated with the risk factors) and the sum of expected VaR (simulated with the risk factors) and the sum of expected loss and concentration riskloss and concentration risk

Generally specific risk and concentration risk can be diversified by Generally specific risk and concentration risk can be diversified by diminishing exposures on some counterparty/industries, while diminishing exposures on some counterparty/industries, while systemic risk cannot be diversified because it’s embedded in the systemic risk cannot be diversified because it’s embedded in the economic systemeconomic system

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Here is illustrated an example of decomposition of Credit VaRHere is illustrated an example of decomposition of Credit VaR

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Decomposition of Credit VaRDecomposition of Credit VaR

0 200 400 600 800 1000 12000

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Losses (mln €)

Simulated portfolio - no correlation

1. Simulation of uncorrelated defaults, we have an Expected Loss of 44,6 millions € and a VaR (99%) of 587,1 millions €. The concentration risk is therefore 542,5 millions €

Expected Loss VaR (99%)

0 500 1000 1500 2000 2500

2%

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6%

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10%

12%

14%

16%

Losses (mln €)

Simulated portfolio - with correlations

1. Simulation of correlated defaults, we have an Expected Loss of 44,6 millions € and a VaR (99%) of 921 millions €. The concentration risk is therefore 378,5 millions €

Expected Loss

VaR (99%)

The economic capitalThe economic capital Generally the confidence level of the Credit Var is given by the Generally the confidence level of the Credit Var is given by the

“target rating” of the bank, i.e. its “target rating” of the bank, i.e. its risk appetiterisk appetite

E.g. if a company wishes to be a “BBB+”, it embeds a PD of E.g. if a company wishes to be a “BBB+”, it embeds a PD of (approximately) 0,50%(approximately) 0,50% the Credit VaR confidence level should be the Credit VaR confidence level should be 99,50%99,50%

Given that, the top management should choose a “lowest tolerable Given that, the top management should choose a “lowest tolerable result” (LTR), that represents the maximum profit erosion that could result” (LTR), that represents the maximum profit erosion that could be “tolerable” for the shareholdersbe “tolerable” for the shareholders

The difference between the Credit VaR and the LTR is the The difference between the Credit VaR and the LTR is the economic capitaleconomic capital

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Let’s suppose:Let’s suppose: A portfolio with1,000 counterpartiesA portfolio with1,000 counterparties Credit VaR (99.5%): 275 millions €Credit VaR (99.5%): 275 millions € Net Income: 135 millions €Net Income: 135 millions € Lowest tolerable result: 70 millions €Lowest tolerable result: 70 millions € Net assets: 150 millions €Net assets: 150 millions €

We simulate the Credit VaR with 1 millions iterations We simulate the Credit VaR with 1 millions iterations (for simplicity, we don’t consider correlations with risk (for simplicity, we don’t consider correlations with risk factor)factor)

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The economic capital – an The economic capital – an exampleexample

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The economic capital – an exampleThe economic capital – an example

50 100 150 200 250 300 350 4000

0.5%

1%

1.5%

2%

2.5%

3%

3.5%

4%

Perdite (milioni di €)

frequenza

Patrimonio netto

Capitale necessario a coprire le perdite

Iniezione di knecessaria

Lowesttolerableresult

UtiliVaR 99.5%

Distribuzione perdite su portafoglio simulato

In this case, net assets and (tolerable) profit erosion are not enough to cover unexpected losses: the company need a capital injection of (at least) 55 millions €

Therefore, the economic capital represents the “real” amount of Therefore, the economic capital represents the “real” amount of capital that a company must have to tackle unexpected eventscapital that a company must have to tackle unexpected events

The concept of economic capital is, in fact, the ultimate goal of The concept of economic capital is, in fact, the ultimate goal of Basel’s II second pillar (for credit risk), because it takes into account Basel’s II second pillar (for credit risk), because it takes into account also concentration risk and systemic risk (with several risk factors)also concentration risk and systemic risk (with several risk factors)

A (desirable) complete model of economic capital should anyway A (desirable) complete model of economic capital should anyway take into account also other risks (e.g. market risk and operational take into account also other risks (e.g. market risk and operational risks) in order to cover all banks’ risksrisks) in order to cover all banks’ risks

How to estimate correlations between credit and operational risks?How to estimate correlations between credit and operational risks? How to integrate different models used to estimate different risks? How to integrate different models used to estimate different risks?

(e.g. parametric market VaR with simulated Credit VaR)(e.g. parametric market VaR with simulated Credit VaR)

The discussion about these issues is very actual not only in banking The discussion about these issues is very actual not only in banking industry, but also for insurance companiesindustry, but also for insurance companies

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The economic capitalThe economic capital

BUT

ReferencesReferences1. Long (1997), Regression models of categorical and limited regression.

2. Resti, A. and Sironi, A., “Risk Management and Shareholders’ Value in Banking”. 2007

3. Standard & Poor’s (2008) 2008 Annual Global Corporate Default Study And Rating Transitions. www2.standardandpoors.com

4. Studies on credit risk concentration (2006). Basel committee on banking supervision. www.bis.org

5. Zhang L., Zhu F., Lee J. (2008), Asset Correlation, Realized default correlation and portfolio credit risk. Moody’s KMV

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